Discontinuous Galerkin and the Crouzeix-Raviart element : application to elasticity
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 1, pp. 63-72.

We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix-Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.

DOI: 10.1051/m2an:2003020
Classification: 65N30,  74B05
Keywords: Crouzeix-Raviart element, Nitsche's method, discontinuous Galerkin, incompressible elasticity
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Hansbo, Peter; Larson, Mats G. Discontinuous Galerkin and the Crouzeix-Raviart element : application to elasticity. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 1, pp. 63-72. doi : 10.1051/m2an:2003020. http://www.numdam.org/articles/10.1051/m2an:2003020/

[1] D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742-760. | Zbl

[2] G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31 (1977) 45-59. | Zbl

[3] S.C. Brenner and L. Sung, Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992) 321-338. | Zbl

[4] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer (1997). | MR | Zbl

[5] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Sér. Rouge 7 (1973) 33-75. | Numdam | Zbl

[6] R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity. Math. Comp. 57 (1991) 529-550. | Zbl

[7] M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles. Internat. J. Numer. Methods Engrg. 19 (1983) 505-520. | Zbl

[8] P. Hansbo and M.G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1895-1908. | Zbl

[9] P. Hansbo and M.G. Larson, A simple nonconforming bilinear element for the elasticity problem. Trends in Computational Structural Mechanics, W.A. Wall et al. Eds., CIMNE (2001) 317-327.

[10] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, New Jersey (1987). | MR | Zbl

[11] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9-15. | Zbl

[12] R. Rannacher and S. Turek, A simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equations 8 (1992) 97-111. | Zbl

[13] F. Thomasset, Implementation of Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, New York (1981). | MR | Zbl

[14] M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152-161. | Zbl

[15] B. Cockburn, K.E. Karniadakis and C.-W. Shu Eds., Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes Comput. Sci. Eng., Springer Verlag (1999). | MR | Zbl

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